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Two step (spherical) square well potential

  • #1

Homework Statement


Just need some rough guidance on this one, nothing specific is really needed. The problem:

Given a spherically symmetric potential (V(r))
[tex]
V\left( r \right) = \left\{\begin{gathered}
V_0 \hfill \hspace{2}r<a \\
0 \qquad a<r<b \\
\infty \hfill r>b \\
\end{gathered} \right
[/tex]

Find the energies for the ground state and the first excited state. Also find an (approximate) expression for the energy splitting of the levels if [itex]V_0[/itex] is very large compared to these energy levels.


Homework Equations



Must use spherical Bessel and Neumann functions.

The Attempt at a Solution



The wavefunctions:

[tex]\mathcal{U}_I(r)=A_rJ_{\ell}(k_1r)[/tex]
[tex]\mathcal{U}_{II}(r)=C_rJ_{\ell}(k_2r)+D_r n_{\ell}(k_2r)[/tex]
[tex]\mathcal{U}_{III}(r)=0[/tex]

[tex]k_I=\frac{\sqrt{2m(V_0-E)}}{\hbar}[/tex]

[tex]k_{II}=\frac{\sqrt{2mE}}{\hbar}[/tex]

where the spherical Neumann function ([tex]n_{\ell}[/tex]) goes away in the first function since it is singular at the origin.

At this point work with the following boundary conditions:

[tex]\mathcal{U}_I(a)=\mathcal{U}_{II}(a)[/tex]
[tex]\mathcal{U}_I^{'}(a)=\mathcal{U}_{II}^{'}(a)[/tex]
[tex]\mathcal{U}_II(b)=0[/tex]

From here you get a three equations, where the derivatives are taken with respect to K1 and K2 (respectfully). First question: How do I take a derivative of the spherical Bessel and Neumann functions? Should I use the recursion formula and solve for the derivative?

Actually I am just not quite sure where to go from here in terms of finding the ground state energy. If this were a regular potential (two step) barrier, all I would do is eliminate the three constants using the three equations, arriving at a transcendental equation which I would then graph graph each side of the equation separately and find the intersections which represent the energy eigenvalues. Same thing here? Any thoughts???

Thanks yall

IHateMayonnaise
 

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